Remark 4.45 
If (QiS) and (UU) hold, we claim that the Quotienting-in-Stages stable locus \(X^{P-qs}\)  is open and non-empty. Indeed, as \(\dim \operatorname{Stab}_{U^{[i]}}(-)\)  is constant on \(X_{\min }^{[i],ss}\) , then \(U^{[i]}Z^{[i],(s)s}_{\min } \subset X_{\min }^{[i],(s)s}\)  is a closed subset by [1]}. Note that \(X^{(s)s}_{\min } \subset X_{\min }^{[i],(s)s}\)  by Lemma REF  (v). Hence, \(X^{P-qs}\)  is open in \(X^{s}_{\min }\) , which itself is open in \(X_{\min }\) . Thus provided \(X_{\min }\)  is open in \(X\) , which for example will be the case if \(X\)  is irreducibleSee \(§\)REF  for how to remove this irreducibility assumption., the Quotienting-in-Stages stable locus \(X^{P-qs}\)  is open in \(X\) . Finally, non-degeneracy (REF ) of (QiS) implies this set is non-empty.

